Equality in Sperner theorem
= Equality in Sperner theorem
{title2=$|\mathcal A|=\binom n{\lfloor n/2\rfloor}$}
The maximum <antichains> in the <Boolean lattice> are exactly its full middle levels. For even $n$ there is one such level, and for odd $n$ either middle level works. The <LYM inequality> first confines a maximum antichain to those levels; connectedness of the regular inclusion <bipartite graph> rules out a mixture of the two odd middle levels.